Spinon Fractionalization from Dynamic Structure Factor of Spin-1

arXiv:1804.03333v1 [cond-mat.str-el] 10 Apr 2018 aoeatfroant(AM ihnearest-neighbor intensively investigated with been has (KAFM) interaction Heisenberg antiferromagnet kagome rbsaehgl desired. experimenta highly to are related probes measurements dynamic on studies cal eoac NR [ [ (NMR) INS resonance experimen- the in including unresolved probes remains gap tal spin finite spin-liquid a a has such didate whether distinguis addition, In for essential physics. different INS the of understanding of oretical continuum similar a disorder to as lead such also material in I factors measurements. other such some from particular, fractionalizati extracted the be and can excitations state, spin the of rega order information topological what ing issue open an remains It mag conventional excitations. of spectrum the from different distinctly (DSSF) factor structure spin character dynamic the measurement the (INS) Furthermore, scattering neutron scale. [ inelastic energy orders coupling few below a magnitude temperatures of to down mater excluded the been of has order magnetic The candidates. spin-liquid ing [ kagome- herbertsmithite ferromagnet as [ such compounds materials triangular-lattice and been magnetic have frustrated candidates in QSL identified Experimentally, experiments. in ain [ exc nature tations fractionalized intrinsic and entanglement the massive shown including have QSL studies Theoretical temperatu limit. zero approaching even breaking symmetry lattice otnu pcrmi ihrfeunyrgm [ regime frequency higher in spectrum continuum ei tts[ mag- states conventional from netic differently behaves which phase tum pnnFatoaiainfo yai tutr atrof Factor Structure Dynamic from Fractionalization Spinon ntertclsuy h rudsaeo h spin- the of state ground the study, theoretical In Introduction.— 4 – 7 ,wihaecalnigt emaue directly measured be to challenging are which ], 1 1 ASnmes 51.m 54.g 75.40.Gb 75.40.Mg, 75.10.Jm, spi numbers: the PACS identify we phase, ordered magnetical transition. the D and the following CSL By i the excitations. and spinon herbertsmithite, fractionalized of with measurements Thes scattering liquid. spin neutron d gapless a DSSF with the consistent more interaction, com conditions Dzyaloshinkii-Moriya intensity weak spectrum a reduced adding mom much from with identified behavior is critical spectrum momenta excitation for spinon region Secondly, frequency high at distribution spectral nest rdmnnl tlwfeunyrgo around region frequency ph low behaviors. at the response predominantly in intensity dynamical nearby distinct sitting liquid shows B spin interaction chiral group. and renormalization state density-matrix ordered of means by (KAFM) hoeia iiin - n NS o lmsNtoa Lab National Alamos Los CNLS, and T-4 Division, Theoretical – 3 3 esuydnmclsi tutr atr(SF of (DSSF) factor structure spin dynamical study We .I osntso n antcodror order magnetic any show not does It ]. eateto hsc n srnm,Clfri tt Univ State California Astronomy, and Physics of Department unu pnlqi QL sanvlquan- novel a is (QSL) liquid spin Quantum 11 .T lrf hs usin,theoreti- questions, these clarify To ]. 8 – 10 2 11 eateto hsc,BiagUiest,Biig 10019 Beijing, University, Beihang Physics, of Department n h ula magnetic nuclear the and ] soeo h otpromis- most the of one is ] S 8 – ( Q 15 ω , .Tekgm anti- kagome The ]. .Zhu W. S ) ( [ Q 16 ω , ,mkn the- making ], 1 10 nteKgm Lattice Kagome the on ) huSuGong Shou-Shu , sabroad a as ,wihis which ], 9 nof on , [ hing can- may izes 17 1 non 10 rd- ial / of re i- – n 2 ] l Q S pngpi tl ne eae hl ale est ma- density [ earlier a liquid While suggested spin simulation gapped debate. (DMRG) under group na- renormalization still the trix is including gap existe QSL the spin the and a quasi-particles, of fractionalized nature the full of established ture been the has KAFM, state the ground in QSL a Although smithite. eetqatmpae,icuiga including for phases, DSSF quantum cou the ferent of pertubative features these characteristic identify With we plings, material. experimental re to both evant are which models interaction, extended (DM) interactio Dzyaloshinskii-Moriya Heisenberg and further-neighbor KAFM small the either with for DMRG large-scale on MGtreigtesse epnet h netdflxand [ flux results inserted network the to tensor re response by system supported the indirectly targeting also DMRG is scenario liquid spin less isdnmrclmdlcluaini aelmtdt small to limited the rare on is [ based systems calculation are model KAFM numerical the [ biased most methods of approximate far DSSF or So the analysis mean-field on properties. studies such excitation the reveals study of that state DSSF ground the the as new beyond demands phase approaches open QSL The the theoretical of absent. nature still the regarding are question excitations energy low from dence [ liquid state spin ground Dirac optimized U(1) the gapless the found as study Carlo Monte tional 31 o.I h S hs,teeeg cn fteDS show DSSF the of scans energy the phase, CSL the ve wave In magnetic corresponding tor. the at intensity wi modes largest dispersive the gapless sharp of appearance the is order n otepaeo h ueKF w eoei sKL.In KSL). as it denote (we KAFM pure the connect- the of QSL a phase and the (CSL), to liquid ing spin chiral gapped a phase, eut atr h anosrain nteinelastic the in observations main the capture results e S rsigteqatmpaetasto between transition phase quantum the crossing SSF = ln h onayo h xeddBilunzone. Brillouin extended the of boundary the along s iga,teKF ihtenearest-neighbor the with KAFM the diagram, ase dct h pnlqi aueo h rudstate ground the of nature liquid spin the ndicate is fal h SFdsly motn spectral important displays DSSF the all, of First 2 ,wihcpue h oiatitrcinfrteherbert- the for interaction dominant the captures which ], nti ae,w i oudrtn h SFbased DSSF the understand to aim we paper, this In 1 = , aigt h egbrn hrlsi iud By liquid. spin chiral neighboring the to paring q 3 M nu n nryrsle SF hc shows which DSSF, resolved energy and entum mntae togsniiiyt h boundary the to sensitivity strong a emonstrates .N Sheng N. D. , rtr,LsAao,NwMxc 74,USA 87545, Mexico New Alamos, Los oratory, (0 = oprsnwt h eldfie magnetic well-defined the with comparison y o odnaindiigteqatmphase quantum the driving condensation non on nmmnu pc,adsosabroad a shows and space, momentum in point / riy otrde aiona930 USA 91330, California Northridge, ersity, 2 39 esnegmdlo h aoelattice kagome the on model Heisenberg , 0) ]. Spin- hs,tekysgaueo ogrnemagnetic long-range of signature key the phase, 1 3 / 24 2 ,China 1, – esnegAntiferromagnet Heisenberg 32 27 , ,pro osrcinadvaria- and construction parton ], 33 .Hwvr oedrc evi- direct more However, ]. q (0 = 19 – , 22 0) 34 .Sc gap- a Such ]. antcorder magnetic – 38 ,teun- the ], c of nce sor ns cent dif- c- th l- - 2

(a) (d)

(b) (c)

S(q ,q ) CSL x y

2.400

0.2 2.300

VBC 2.200

2.100

2.000

1.900

1.800 3

1.700 J

1.600

1.500

1.400

1.300

1.200

1.100

0.1 1.000

0.9000

0.8000

0.7000

0.6000

0.5000

0.4000

0.3000

0.2000

0.1000 KSL q = (0,0) 0.000 0 0 0.05 0.1 0.15 0.2 0.25 J2

FIG. 1. Static spin structure factor of the kagome model in different quantum phases. (a) Quantum phase diagram of the kagome model in the J2 − J3 plane obtained in Ref. [40]. (b-d) are static spin structure factor in momentum space for (b) the q = (0, 0) phase at J2 = 0.25, J3 = 0.0, (c) the CSL at J2 = 0.25, J3 = 0.25, and (d) the KSL at J2 = J3 = 0. The extended Brillouin zone is marked by the white dashed line. intensity peak at finite frequency, which illustrates the emer- as gent gapped spinon pair excitations. In the KSL, the momen- z z 1 iQ·(ri−rj ) z z tum resolved DSSF concentrates along the boundary of the S(Q)= hS (−Q)S (Q)i = e hSi Sj i, N X extended Brillouin zone (BZ) and shows a broad maximum at i,j the M point, which are consistent with the INS results of the where the wave vector Q ~ ~ in the herbertsmithite. In the energy scans for the KSL, the intensity = (q1, q2) = q1b1 + q2b2 ~ of the DSSF forms a continuum, which extends over a wide BZ is defined by reciprocal vectors b1,2 (see Fig. 2(d)). In frequency range, concomitant with a pronounced intensity at Fig. 1(b), S(Q) shows sharp peaks at the M points, showing low energy region. The evidences from DSSF, including the a q = (0, 0) magnetic order [45]. In the nonmagnetic phases fractionalized spinon continuum in energy scans, the sensitiv- S(Q) is featureless as shown in Fig. 1(c-d). In the KSL phase, ity of excitation gap by imposing different boundary condi- S(Q) concentrates along the boundary of the extended BZ tions (BCs), and by tuning DM perturbation, are in support of and shows broad maximum near the M point, which agree a QSL with gapless fractionalized spin excitations. with the features of the INS data of herbertsmithite [10]. Dynamic spin structure factor.— The DSSF is defined as Model and Method.— We study the spin-1/2 KAFM with further-neighbor antiferromagnetic interactions 1 1 Sαβ(Q,ω)= − ImhSα(−Q) Sβ(Q)i, π ω − (H − E0)+ iη

H = J1 X Si ·Sj +J2 X Si ·Sj +J3 X Si ·Sj , (1) where E0 is the ground-state energy, η → 0 is a small smear- hi,ji hhi,jii hhhi,jiii ing energy [44], and α, β denote spin components. First of all, we discuss the salient features of the DSSF in different phases as shown in Fig. 2. For the q phase (Fig. 2(a,d)), where J ,J ,J are the first-, second-, and third-neighbor = (0, 0) 1 2 3 we observe a sharp peak at the point with , serving couplings (J is the coupling inside the hexagon and we take M ω = 0 3 as the key signature of the long-range magnetic order with J1 =1 as the energy scale). The previously obtained DMRG phase diagram is shown in Fig. 1(a) [40]. Different neighbor the largest intensity at ordering wave vector. Interestingly, we also observe a small peak with a broader and reduced phases surround the KSL sitting near the J point, including a 1 weight in higher energy region, which we speculate related to q = (0, 0) magnetic order phase, a gapped CSL phase, and a valence-bond solid phase. two-magnon excitations. For all other momenta, the intensity shows broad distribution in the energy scans. In this study, we develop a DMRG program to calculate In the CSL, the DSSF along the high-symmetricline is pre- dynamic structure factor [41–43], which can apply to general sented in Fig. 2(b,e), showing a fully gapped excitation branch strongly correlated systems. We consider cylinder geometry at the M point. The extracted spin gap 0.4J1 is consistent with closed boundary in the y direction and open boundary in with a direct measurement of the gap in static simulation. For the x direction, with the number of sites N = 3 × Lx × Ly other momentum points along the boundary of the extended (Lx ≫ Ly), where Lx and Ly are the numbers of unit cells BZ, the DSSF has broad distribution with suppressed inten- along the x and y directions, respectively. We first obtain the sity supporting the spin spectrum as a convolution of the frac- ground state of a long cylinder, and then target the dynami- tionalized excitations. Since theoretically the CSL is well de- cal properties by sweeping the middle Ly × Ly unit cells to scribed as the Laughlin state with spinons satisfying semionic avoid edge excitations (see Sec. of [44] for details). Most of statistics [46], the intensity peak at the M point should be calculations are performed on the Ly = 4 cylinder. For the composed of spinon pair excitations (see Ref. [44]). q = (0, 0) phase, we obtain well converged results also for Next we turn to the KSL as shown in Fig. 2(c,f). The dom- Ly =6. inant intensity of the DSSF is also carried by the momentum We first present the static spin structure factor that is defined near the M point, and the spectrum at each momentum shows 3

(a) Neel phase, J =0.25, J =0 (b) CSL, J =J =0.25 (c) KSL, J =J =0

2 3 2 3 2 3

1.0 1.2 1.2

6.000

0.9400 1.085

5.600

0.8600

0.9600

4.850

1.0 1.0

0.8

0.7100 4.100

0.7600

3.350

0.8 0.8

0.5600

0.6 2.600

0.4100

1.850 0.6 0.6 0.3600

1.100

0.2600

0.4

0.1600

0.3500 0.4 0.4

0.1100

0.000

-0.4000 0.000 -0.04000

0.2

0.000

-0.04000 0.2 0.2

0.0 0.0 0.0

M M M K M K M 2 M 2 K 1 1

2 1

(d)

(e) (f)

FIG. 2. Dynamic spin structure factor in different quantum phases. (a-c) Contour plots of the DSSF as a function of energy and momentum for (a) q = (0, 0) phase at J2 = 0.25, J3 = 0.0, (b) CSL at J2 = 0.25, J3 = 0.25, and (c) KSL at J2 = J3 = 0. The white and black dashed line in (c) shows the constant energy scan at low-frequency and high-frequency region, respectively, which can be compared with the INS observations in herbertsmithite (see [44] for details). (d-f) The energy scans of the DSSF with the momentum along the path Γ → M1 → K → M2 in extended BZ. The intensity scales differ among the different panels. The inset of (d) shows reciprocal vectors of kagome lattice in the Brillouin zone with denoted high-symmetry momentum points. broad distribution and spans a wide energy region. For ex- region is similar to the case of one dimensional Heisenberg ample, S(M,ω) shows a dominant intensity at small energy model where a critical spin liquid phase has been identified as and a long tail up to ω ≈ 1.2; the overall feature is quite dif- the ground state with gapless spinon excitations [47]. We re- ferent from the spectrum of the q = (0, 0) phase, where the mark that the DSSF results in the KSL phase capture the main overwhelming part of the spectral weight is carried by energy features of the INS results of herbertsmithite, including the ω = 0. Compared with the CSL phase, here the spectrum low-energy spectrum peak at the M point and the flat spin ex- weight moves down in energy, consistent with a reduction citations between the M and K points at higher energy, which of spin excitation gap. Interestingly, for the CSL phase, al- would be discussed below in detail. though the DSSF forms a continuum along the extended BZ Spinon condensation and quantum phase transition.— It is boundary,the energyscan of dynamicalspin structure factor at also interesting to study the quantum phase transition from the each momentum point shows a dominate peak structure with view of DSSF, which reveals the dynamic driving mechanism narrow broadening width, which supports a deconfined sta- of the transition. Here we study the transition from the CSL ble (long-lift time) spinon excitations. Comparing with these to the q = (0, 0) phase (see Ref. [44]). From the evolution characteristic features of the CSL, the picture shown in Fig. of the DSSF at the M point by tuning J3, we observe the 2(d,h) for the KSL indicates such a spectrum is still related to following key features: In the CSL phase (J3 > 0.18), with fractionalized spinon excitations with much reduced life time. the system approaching the transition point, the predominant The appearance of excitation continuum in high frequency peak moves towards the low-frequency regime and the peak intensity gradually increases. After entering the Néel phase (J < 0.18), the predominant peak appears exactly at ω = 0, 3

8 8 8 8

J =0.05 which is well separated from the high-frequency excitations. J =0.25 J =0.15 J =0.20

3 3 3 3

Neel phase CSL phase Neel phase CSL phase

6 6 6 6 The above observations indicate that the quantum phase tran-

) sition between the CSL and the N´eel phase can be understood

4 4 4 4 as driven by the condense of the spinon pairs to form the spin-

2 2 2 2 1 magnon excitations. S(Q=M, Connection with experiment.— In Fig. 4, we show the plots

0 0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 of the DSSF at constant energy, and compare our results qual- itatively to the experimental data (Fig. 1 in Ref. [10]). The FIG. 3. Evolution of dynamical spin structure factor at Q = M main observation from INS experiment is that, in the low fre- point, by varying J3. By decreasing J3 from ﬁnite to zero, It is ex- quency region the measured DSSF shows the peak structure pected that chiral spin liquid undergoes a continuous phase transition around the M points; while in the higher frequencies, the to the q = (0, 0) phase [40]. peak structure is smeared out, and the DSSF is almost ﬂat dis- 4

(b) (a)

0.20

(b)

Periodic

Anti-Periodic (pi-flux)

2 / 3

2 / 3 0.15

J =0

2 s

0.10

4 /3 -4 /3

-4 /3 4 /3

0.05 SpinGap

2 / 3 2 / 3

0.00

0.02 0.04 0.06 0.08 0.10 0.12

z y

D q q

-4 /3 4 /3

-4 /3 4 /3 (c) Periodic BC (d) Anti-Periodic BC q 2.5 2.5

x q

x zz zz

S (Q=M, S (Q=M, )

2.0 2.0 +-

+- S (Q=M, )

S (Q=M, )

FIG. 4. Comparison between experimental measurements and nu- 1.5 1.5

merical results for the DSSF. (Top) Experimental data at ﬁxed fre- 1.0 1.0 quency are shown for (left) and (right) ω = 0.75meV ω = 6meV 0.5 0.5 (The data are from experimental group [10]). (Bottom)Theoretical

0.0 0.0 results for DSSF at fixed frequency are plotted for ω = 0.2 ≈ 3meV 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (left) and ω = 0.6 ≈ 10meV (right). The extended Brillouin zone is indicated by the white dashed line. FIG. 5. (a) Phase diagram of KAFM by including second nearest- z neighbor coupling J2 and out-of-plane DM interaction D . The tributed along the boundary of the extended BZ [10]. Here we squared dot represents the KAFM with nearest-neighbor couplings. Red star line shows the possible parameter regime for herbert- re-present two constant energy plots of the DSSF from exper- z imental measurements in low frequency (ω = 0.75mev) and smithite [49, 50]. (b) Spin gap for various D under periodic (blue squares) and anti-periodic (purple dots) BC. The DSSF of KSL phase high frequency ω =6meV, respectively as shown in Fig. 4(a- z at J2 = 0, D = 0.06 under (c) periodic BC and (d) anti-periodic b). Accordingly, we show two calculated DSSF plots at two BC, for longitudinal mode (purple dashed line) and transverse mode constant energies in Fig. 4(c-d). Our numerical DSSF devel- (blue line). ops peak structures around the M points in low frequency and flat distribution along the boundary of the extended BZ in high frequency, respectively. Through this comparison, we into higher energy region. Importantly, the low-energy excita- conclude that the fractionalized spinon spectrum obtained in tions are governed by the transverse mode, which also shows calculations can capture the main experimental observations, substantial difference by tuning BC. The dominated spectral both in the low frequency and the high frequency regime. peak in the anti-periodic BC shifts to zero frequency, showing While the KAFM is generally believed as a good start- gapless spin excitations. These results are consistent with the ing point to understand the spin-liquid-like behaviors of her- KSL as a critical phase. bertsmithite, the spin-orbit coupling in the absence of inver- Summary.— We have studied the DSSF of the spin-1/2 sion symmetry between two adjacent Cu2+ irons yields a DM Heisenberg model on the kagome lattice with either further- interaction Dij · (Si × Sj ) [48] in herbertsmithite. Elec- neighbor or additional DM interactions using DMRG. The tron spin resonance [49] and magnetic susceptibility measure- DSSF of the kagome spin liquid shows different characteri- z ments [50] suggest an out-of-plane DM interaction Dij ≈ zations from those in the gapped CSL or in the q = (0, 0) 0.04 ∼ 0.08J1. To make a bridge between experiments and magnetic phase, which concentrates along the boundary of numerical simulations, we study the DSSF of the KAFM with the extended Brillouin zone with broad maximum near the additional DM interaction. M point. In the energy scans, the dominant intensity shifts First of all, we show the phase diagram of the system as to low-energy region, and a wide spectral distribution spans to z a function of D and J2 in Fig. 5(a) (we set J3 = 0), in- high-energy region, showing a continuum expected for a spin cluding the KSL and q = (0, 0) phase. We obtain the phase liquid state. Besides, the DSSF captures the main features diagram by studying the magnetic order parameter [44]. In of the inelastic neutron scattering features of herbertsmithite. z We also propose that the DSSF could be used to characterize the absence of J2, we find a transition at Dc ≈ 0.08, slightly smaller than previous result [34, 51, 52]. With increasing Dz, exotic quantum phase transitions. the spin-1 excitation gap decreases monotonically as shown in Note added.— In the stage of finalizing our paper, we be- Fig. 5(b). For Dz < 0.08, the spin gap depends on the BCs, came aware of new preprints [53, 54], which study the dynam- similar to the DMRG results of the pure kagome model. Since ical properties of the Z2 spin liquid in a sign free anisotropic DM interaction breaks spin rotational symmetry, we calculate kagome model by the quantum Monte Carlo method. the DSSF in both the longitudinal and transverse modes as Acknowledgments.— W.Z. thanks for C. D. Batista, S. S. shown in Fig. 5(c-d), under different BCs. It is found that the Zhang, Z. T. Wang and Y. C. He for insightful discussion. intensity distribution of the DSSF remains similar to the re- W.Z. also thanks T. Han for discussing experimental data. sults of the KAFM, showing broad distribution and long tail This work was supported by the U.S. Department of Energy 5

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Numerical Method

In this section, we introduce the numerical simulation details about dynamical properties in density-matrix renormalization group (DMRG) algorithm. We also provide a benchmark on square Heisenberg model to show the high accuracy of DMRG algorithm.

1. Density-matrix renormalization group algorithm

We perform the calculations based on high accuracy DMRG on cylinder geometry with closed boundary in the y direction and open boundary in the x direction. We denote it as Ly × Lx (Lx ≫ Ly), where Ly and Lx are the number of unit cells in the y and x directions. We first perform the ground state DMRG procedure and sweep the ground state on the whole cylinder, and then target the dynamical properties (see below) by sweeping the middle Ly × Ly unit cells to avoid edge excitations. Most of the calculations are performed on Ly =4 and Ly =6 cylinders. Here we would like to point out that, although the whole system on cylinder does not host translational symmetry along x-direction, the ground state in the middle of a long cylinder approximatelly satisfies the translational symmetry (the emergent translational period determined by the nature of the ground state itself). Due to this reason, we can cut the middle Ly × Ly system and glue it into a torus (with periodic boundary condition along both x- and y-direction), so that the momentum quantum number can be well defined along both x- and y-direction (within Ly × Ly unit cells in the middle of the cylinder). This process is widely used for spin structure factor calculations in DMRG community. The conventional DMRG algorithm only targets the ground state, |0i. To calculate dynamical spin structure factor, we apply the dynamical DMRG by targeting the following states together with the ground state when sweeping:

|Sα(Q)i = Sα(Q) |0i |xα(ω + iη)i = 1 |Sα(Q)i ω+iη−(H−EGS ) where |x(ω)i is usually called correction vector which can be calculated by the conjugategradientmethod [42] or other algorithm [43]. With the help of the correction vector, the dynamical spin structure factor can be calculated directly:

1 Sαβ(Q,ω)= − ImhSα(Q)|xβ(ω + iη)i (2) π where η takes a small positive value as the smearing energy. Taking these states (|0i,|Sα(Q)i and |xα(ω + iη)i) as target states and optimizing the DMRG basis to represent them allow for a precise calculation of the structure factor for a given frequency ω and the broadening factor η. In this work, all calculations are performed using η =0.05 and η =0.1 (in unit of nearest-neighbor coupling J1). Since we have to target multi-states in the DMRG process, the truncation error is basically larger than the ground state DMRG. In this work, we ensure the truncation error of the order or smaller than 10−5, by keeping up to 2400 states. Here we also comment on the numerical scheme we used in this paper. In general, there are two main algorithms to target dynamicsbased on DMRG algorithm. One is to calculate the dynamical spin structure factor in the frequency regime (as outlined above), the other one is to first calculate the time-evolution of the physical quantities and then obtain the dynamical spin structure factor by Fourier transformation. In general, the first method is more accurate in the low-frequency regime, while the second method works better in high-frequency regime (because the accumulated errors grow as time steps increases in time-dependent DMRG). Based on this reason, in the discussion of low-energy physics of kagome Heisenberg model, we utilize the first method ([42, 43]) in this paper.

2. A benchmark: Neel order on square lattice

In this section, as a benchmark of DMRG method, we show the dynamical spin structure factor of the S =1/2 antiferromagnetic Heisenberg model on the square lattice. For J1 Heisenberg model on square lattice, the ground state is a q = (π, π) Neel ordered state. Thus, we expect to see a single-mode gapless excitation dispersion related to magnon quasiparticle in dynamical spin structure factor [? ]. Fig. 6 shows the energy-dependence of the dynamical spin structure factor S(Q,ω) at several typical momenta. We further extract the peak position at each momentum point and plot the single magnon dispersion in Fig. 7, which is in largely agreement with the spin wave theory (For q = (π, π) Neel order, it is believed that spin wave theory can capture the main features of dynamics except for Q = M point.). The dominant peak is a single magnon excitation. Importantly, the largest weight is carried by S(Q = X,ω = 0), and the peak location of Q = X = (π, π) centered at ω =0 directly reveals the gapless nature for Neel q = (π, π) phase. Interestingly, the S(Q = M,ω) shows an anomalous tail in high energy regime, which 8 could be attributed to the fact that magnon-magnon interaction is enhanced near Q = M [? ]. The discrepancy near the M point between our result and the linear spin wave theory comes from that a linear spin wave theory neglects the magnon-magnon interactions (with ﬁrst-order corrections, it is known that single magnon dispersion shifts upward). Here, through this benchmark on the square lattice and extensive tests on one dimensional chain (not shown here), we conclude that the current scheme can obtain reliable dynamical properties efﬁciently. Next we will apply the above strategy to antiferromagnetic Heisenberg model on the kagome lattice.

1.0 1.0

Q=(0,0)= Q=( , )=S

0.8 0.8

0.6 0.6 )

0.4 0.4 S(Q,

0.2 0.2

0.0 0.0

0 1 2 3 0 1 2 3

/J /J

20 1.0

Q=( , )=M Q=( , )=X

0.8

0.6 )

0.4 S(Q,

0.2

0 0.0

0 1 2 3 0 1 2 3

/J /J

FIG. 6. The energy scan of dynamical spin structural factor S(Q,ω) for Heisenberg model on the square lattice, for several typical momentum points in BZ. The calculations are performed on Ly = 8 cylinder by keeping M = 400 states.

DMRG

3.0 0.3 Linear Spin W ave

3.0

2.5

2.0

1.5

1.0

0.5

0.0

X S

FIG. 7. Synthetic the peak position (blue dots) of the longitudinal dynamical structure factors along a path of highly symmetric points in the Brillouin zone. The size of each blue circle is proportional to the static spin structure factor by sum up dynamical spin structure factor over energy. Red solid line shows the dispersions obtained within linear (harmonic) spin-wave theory.

3. Analysis of the Finite-size Effect

In this work, we utilize the DMRG algorithm to simulate the dynamical response. Although we can easily go beyond the exact diagonalization limit, the DMRG calculation still suffers from the finite-size effect, which will be discussed in detail here. Since the cylinder geometry is preferred in DMRG algorithm, the available lattice system sizes are limited by the width along the wrapped direction similar to the ground state DMRG (saying, Ly, which accounts the number of unit cells in the wrapped direction). On the kagome lattice, the current computational ability is limited to accessing Ly up to 6, depending on the nature of different phases. To be specific, the largest system size is Ly =6 for Neel q = (0, 0) order and chiral spin liquid. While for quantum spin liquid the largest available system is Ly =4, because the highly frustrated nature near J1 Heisenberg point leads to the slow convergence in dmrg calculations. We have extensively checked that, the main features of dynamical spin structure factor are robust for Neel q = (0, 0) phase and chiral spin liquid phase, by tuning the system sizes Ly = 4, 5, 6. Thus, we have confidence that the nature of dynamical responses of these two phases shown in the main text is intrinsic properties of the corresponding two-dimensional systems. 9

4 4 4

(a) (b) (c)

3 3 3 )

2 2 2

S(Q,

1 1 1

0 0 0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1.4 1.4 1.4

(e) Pinning BC (f) Anti-Period BC (d) J =J =0 Period BC

2 3

1.2 1.2 1.2

1.0 1.0 1.0 )

0.8 0.8 0.8

0.6 0.6 0.6 S(Q=M,

0.4 0.4 0.4

s 0.2 0.2 s 0.2

0.0 0.0 0.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

J J J

FIG. 8. DSSF for CSL phase (a-c) and KSL phase (d-f) for different boundary conditions (BCs): (a,d) periodic BC in wrapped direction (b,e) periodic BC on wrapped direction and a pinning ﬁeld on the open direction, and (c,f) anti-periodic BC in wrapped direction.

Nevertheless, for the quantum spin liquid phase, we cannot fully rule out the finite-size effect based on Ly =4 system, since Ly =4 is the only available system size. (We cannot reach a converged ground state in dynamical dmrg algorithm for Ly =5, 6 for quantum spin liquid phase due to the difficulty in convergence in such a state). In the main text, we utilize the twisted boundary condition to inspect the gapless nature on a given finite-size system. The main physical reason is further clarified here. First, tuning the boundary condition is a general method to detect the nature of ground state on finite-size calculations. Since the available discrete momentum vectors are limited due to the finite-size effect, tuning the twisted boundary condition allows us to reach more momentum points in the Brilliuin zone. Second, for ground states with intrinsic topological orders, it is expected that the ground state manifold is robust to the twisted boundary condition, without energy level crossing with higher energy levels. In contrast, energy level crossing may occur by tuning boundary conditions if the ground state is gapless. Here the picture is akin to the Thouless’s picture of localization: The energy spectral flow of insulators is robust against boundary conditions, however, energy flow of metallic phase is not. For gapless phase, the change of energy spectrum by twisted boundary conditions inevitably leads to substantial difference in dynamical response functions. Based on these reasons, we inspect the dynamical response for quantum spin liquid phase by tuning different boundary conditions. This is a way out for uncovering the intrinsic nature of ground state on the finite-size calculation.

4. Tuning boundary conditions

We address the question whether or not the ground state of the KSL is gapped, which holds the clue to distinguish the different theoretical scenarios [19][55][? ][56], by inspecting the response of the system under tuning different BCs. As a benchmark, we first test the system in the gapped CSL phase. Since the CSL is equivalent to the ν =1/2 bosonic Laughlin state [57], the system has two-fold topological degenerate ground states. In DMRG simulation, the ground state in the spinon sector can be obtained by adiabatically changing the BC by a 2π phase. As shown in Fig. 8(a-b), the DSSF in the two ground states are almost identical, which can be understood by the fact that local measurements are unable to distinguish different topologically degenerate ground states. Now we inspect the response of the KSL. Fig. 8(d) shows S(Q = M,ω), by imposing periodic BC on the wrapped direction (same with Fig. 2). As a comparison, Fig. 8(e) shows the case of additionally pinning a spinon at each open end of cylinder geometry. Although the spin gap remains robust, the predominant spectral peak in low-energy regime becomes broader. More- over, by imposing anti-periodic BC in wrapped direction, as shown in Fig. 8(f), the excitation gap ∆s shrinks from ∆s ≈ 0.16 to a smaller value ≈ 0.075, signaling that the spin excitation gap is sensitive to the BC. Here, the dramatical change of lineshape of spectral peak and the shrink of spin gap in DSSF, indicate that the ground state is near critical or having very small gap. Of course, the finite size effect is generally important, which calls for future work on finite-size scaling analysis. 10

Transition from chiral spin liquid phase to Neel q = (0, 0) phase

In this section, we study the phase transition from the chiral spin liquid phase to magnetic Neel q = (0, 0) phase. This phase transition is interesting due to the following reasons. First, it is intriguing to understand the low-energy peak structure in dynamical spin structure factor at Q = M point. Second, it is an exotic example of continuous phase transition between gapped topological ordered state and topological trivial state. According to the global phase diagram in the main text (Fig. 1(a)), for finite J2 > 0.15 different phases may appear depending on J3. Tuning J3 will drive a phase transition from chiral spin liquid phase to magnetic q = (0, 0) phase. And it has been found that, chiral spin liquid undergoes a continuous phase transition to Neel q = (0, 0) phase [40], as evidenced by the fact that all local order parameters change smoothly across the phase transition point. However, the reason for this continuous phase transition is less understood before, because the transition between a gapped topological ordered phase and a topological trivial phase is often thought to be first-order type. Next we will show the evolution of dynamical spin structure factor for various J3, by setting J2 =0.25J1. We will focus on momentum wave vector Q = M in this section. As shown in Fig. 3, we show the evolution of dynamical spin structure factor at momentum point Q = M, for various J3. The key features are: 1) In chiral spin liquid phase (J3 > 0.18J1), there exists a peak structure at low frequency regime at Q = M point, as discussed in the main text. By approaching the transition point, this peak structure moves towards the low-frequency regime, and peak intensity gradually increases. 2) In the Neel order phase, the peak position is centered at ω =0. 3) In chiral spin liquid phase, the predominant peak is connected to the high-frequency spin continuum, while in Neel phase the zero-frequency peak is well separated from high-frequency spin excitations. Based on the above observations, a natural interpretation of the peak structure in chiral spin liquid phase is two-spinon resonance state. The reason is that, it is well-known the peak at ω = 0 in Neel phase relates to magnon quasiparticle, which can be viewed as a bound state of two spinons. Taking into account that the element excitation in the chiral spin liquid phase is deconfined spinon, we can take the peak in chiral spin liquid as two-spinon resonance state, while the peak in Neel phase as two- spinon bound state (equivalent to magnon state). Two-spinon resonance naturally depends on the interaction coupling J3. By approaching critical point, two-spinon resonance moves towards zero frequency. In the vicinity of the critical point, two-spinon resonance state becomes two-spinon bound state (equivalent to a magnon). The further condensation of pair spinons should lead to formation of Neel magnetic order. In a word, this picture leads to two important physics: First, the peak of dynamical spin structure factor at momentum Q = M in chiral spin liquid can be interpreted as two-spinon resonance state. Second, the transition from chiral spin liquid to Neel phase can be understood by the formation of condensate of two-spinon bound state or magnon. It therefore provides a microscopic understanding of continuous phase transition between chiral spin liquid and Neel phase. In the above analysis, the peak structure in dynamic spin structure factor of Neel phase and chiral spin liquid occur at the same momentum point (Q = M), which makes the mechanism of spinon pair condensate possible. If the magnetic wave vector of underlying long-ranged magnetic order is different from that of two-spinon resonance state in spin liquid, the phase transition from chiral spin liquid to magnetic ordered phase should be first order. For example, the transition from chiral spin liquid to cuboc1 phase in the global phase diagram is first-order type [40]. To sum up, the evolution of dynamical spin structure factor acrossing the critical point, not only elucidate the nature of the ground state, but also provides invaluable insights on the nature of related phase transition.

Spin correlations and spin gap in the presence of DM interaction

z In the main text, we show a phase diagram as a function of DM interaction D and next-nearest-neighbor coupling J2. The phase boundary between spin liquid phase and magnetic q = (0, 0) phase is determined by the spin gap and spin correlations. z In Fig. 9(a-b), we show the spin gap dependence on parameter J2 and D , respectively. It is found that the spin gap decreases monotonically as approaching the phase boundary. In particular, in the spin liquid phase, the spin gap strongly depends on the twisted boundary condition, indicating the ﬁnite spin gap is due to ﬁnite-size effect. In contrast, in the magnetic ordered phase, the spin gap has little dependence on twisted boundary condition. As shown in Fig. 9(c), the spin correlation provides another z evidence for phase boundary. The spin correlation exponentially decays with the distance, for D < 0.08 and J2 = 0.0. For Dz ≥ 0.08, the long-ranged order emerges as the spin correlation tends to saturate. Based on these facts, we determine the z D ≈ 0.08 as the phase boundary at J2 = 0.0, which is largely consistent with the previous estimation from ED calculation [? ]. Using the similar method, we determine the phase boundary for non-zero J2 case, and map out the full phase diagram as shown in the main text. 11

0.20 0.20 0.1

(a) (b) (c)

Periodic Periodic

Anti-Periodic (pi-flux) Anti-Periodic (pi-flux)

0.01

0.15 0.15 J =0

D =0.03 2

z s s

1E-3

0.10 0.10

1E-4

0.003

0.005

0.05 0.05 SpinGap SpinGap

0.008

1E-5

0.010 Spincorrelations

0.015

0.00 0.00

1E-6

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.02 0.04 0.06 0.08 0.10 0.12

1 2 3 4 5 6 7 8 910

2 Distance (in unit cell)

z z FIG. 9. (a) Spin gap as a function of D by setting J2 = 0 and (b) spin gap as a function of J2 by setting D = 0. Spin gap is obtained by tot tot tot tot ∆s = E0(Sz = 1) − E0(Sz = 0), where the lowest energy state of Sz = 1 is computed by targeting Sz = 1 in the center of the tot + − z cylinder based on the ground state in Sz = 0. (c) spin correlations hSi Si+di for various D , by setting J2 = 0.0. These results are obtained on Ly = 4 cylinder.